In my interpretation, it's unclear
The phrasing of the feature might indicate that they're mutually exclusive ways to use your action
The description of the paladin's Lay on Hands feature says:
Your blessed touch can heal wounds. You have a pool of healing power
that replenishes when you take a long rest. With that pool, you can
restore a total number of hit points equal to your paladin level × 5.
As an action, you can touch a creature and draw power from the pool to
restore a number of hit points to that creature, up to the maximum
amount remaining in your pool.
Alternatively, you can expend 5 hit points from your pool of healing
to cure the target of one disease or neutralize one poison affecting
it. You can cure multiple diseases and neutralize multiple poisons
with a single use of Lay on Hands, expending hit points separately for
The word "alternatively" is the key to determining the answer to your question. The term has no special game meaning, so we must rely on the general English definition. The word "alternatively" is used to link a pair of possibilities/alternatives - and the noun "alternative" refers to: "One of several mutually exclusive things which can be chosen."
The use of the word "alternatively" here seems to suggest that you can either use your action to expend (any amount of) HP from the pool of healing power to restore HP to another creature - or, instead of doing that, you can expend 5 HP from the pool to cure a disease/neutralize a poison. In other words, they seem to be mutually exclusive options.
However, the feature doesn't specifically state the action cost of using Lay on Hands to cure diseases/neutralize poisons, so we rely on the previously stated action economy cost of Lay on Hands: 1 action. As such, we can't entirely ignore the first use of Lay on Hands when attempting to understand how the second use works.
In addition, it does say you can cure multiple diseases and neutralize multiple poisons with "a single use of Lay on Hands" (while expending HP from the pool separately for each), i.e. as part of the same action. Unfortunately, it doesn't specify whether this also extends to restoring HP to a creature as part of the same action - which is the crux of this question.
The feature description doesn't explicitly say that you can both restore a creature's HP and cure disease/neutralize poison as part of the same action. The use of the word "alternatively" could mean that they're mutually exclusive ways to use your action... or it could just mean that they're two different ways to expend HP from the pool (i.e. you expend HP separately for restoring HP vs. curing disease, rather than "double-counting" the expended HP).
In any case, the wording is unclear.
...But Crawford seems to disagree
Rules designer Jeremy Crawford addressed this exact question in an unofficial tweet from November 2016:
So yeah. Lay on Hands. Can you both heal HP *and* cure a disease in one action? I think yes.
In this tweet, Crawford seems to have taken the position that you can indeed restore a creature's HP and cure it of a disease as part of the same action.
However, his one-word response doesn't really clarify his reason for this interpretation. Perhaps the intent is that they're just different ways to expend your HP, but can be done as part of the same action - or perhaps he misread/misunderstood the question. Given this lack of explanation, Crawford's unofficial ruling isn't very strong support for a "yes" answer to this question.
As such, there doesn't seem to be a conclusive answer either way, in my interpretation. My own reading of the feature description still favors a "no" answer.
That said, I don't think it's particularly unbalanced either way, whether you allow or disallow it. I think it's relatively rare that a character is both missing a lot of HP and is affected by an ongoing disease/poison in a scenario where action economy matters enough to need to fix both problems at once. As such, I don't think it matters too much whether you allow or disallow this, at least in most cases.